Skip to main content
Library homepage
 

Text Color

Text Size

 

Margin Size

 

Font Type

Enable Dyslexic Font
Mathematics LibreTexts

0.06e: Exercises - Rational Expressions

  2. Last updated
    Aug 17, 2025
  3. Save as PDF

( \newcommand{\kernel}{\mathrm{null}\,}\)

A: Simplify Rational Expressions.

Exercise 0.6.1 

★ Simplify each rational expression.  Find all numbers that must be excluded from the domain of the simplified rational expression.

  1. 25⁢x95⁢x5
  2. 64⁢x816⁢x3
  3. x2−64x2+16⁢x+64
  4. x2+x−20x2−25
  5. 9−4⁢x22⁢x2−5⁢x+3
  1. x−3⁢x29⁢x2−6⁢x+1
  2. 2⁢x2−8⁢x−422⁢x2+5⁢x−3
  3. 6⁢x2+5⁢x−43⁢x2+x−4
  4. x3+x2−x−1x2+2⁢x+1
  5. 2⁢x3−5⁢x2−8⁢x+202⁢x2−9⁢x+10
  1. 66⁢x⁢(2⁢x−5)18⁢x3⁢(2⁢x−5)2
  2. 26⁢x4⁢(5⁢x+2)320⁢x5⁢(5⁢x+2)
  3. x2+5⁢x+6x2−5⁢x−14
  4. x2−8⁢x+12x2−2⁢x−24
  5. 1−x25⁢x2+x−6
  1. 4−9⁢x23⁢x2−8⁢x+4
  2. 4⁢x2+15⁢x+99−x2
  3. 6⁢x2+13⁢x−525−4⁢x2
  4. x2−5⁢x+4x3−x2−16⁢x+16
  5. x4+4⁢x2x3+3⁢x2+4⁢x+12
Answers to odd exercises:
1. 5⁢x4;x≠0
3. x−8x+8;x≠−8
5. −2⁢x+3x−1;x≠1,32
7. 2⁢(x−7)2⁢x−1;x≠−3,12⁢4 		 9. x−1;x≠−1
11. 113⁢x2⁢(2⁢x−5);x≠0,52
13. x+3x−7;x≠−2,7		 15. −x+15⁢x+6;x≠−65,1
17. −4⁢x+3x−3;x≠±3
19. 1x+4;x≠1,±4

B: Multiply or Divide Rational Expressions.

Exercise 0.6.2 

★ Multiply or divide as indicated, state the restrictions, and simplify.

  1. 14⁢(x+12)25⁢x3⋅45⁢x42⁢(x+12)3
  2. 27⁢x620⁢(x−7)3⋅(x−7)554⁢x7
  3. x2−6436⁢x4⋅12⁢x3x2+4⁢x−32
  4. 50⁢x5x2+6⁢x−27⋅x2−81125⁢x3
  5. 2⁢x2+7⁢x+53⁢x2⋅15⁢x3−30⁢x22⁢x2+x−10
  1. 3⁢x2+14⁢x−52⁢x2+11⁢x+5⋅4⁢x2+4⁢x+16⁢x2+x−1
  2. x2+4⁢x−215⁢x2+10⁢x÷x2−6⁢x+9x2+9⁢x+14
  3. x2−499⁢x2−24⁢x+16÷2⁢x2−13⁢x−76⁢x2−5⁢x−4
  4. 5⁢x2+x−64⁢x2−7⁢x−15÷1−x24⁢x2+9⁢x+5
  5. 6⁢x2−8⁢x−84−9⁢x2÷3⁢x2−4⁢x−49⁢x2+12⁢x+4
  1. x2+4⁢x−12x2−2⁢x−15÷2⁢x2−13⁢x+186⁢x2−31⁢x+5
  2. 8⁢x2+x−925⁢x2−1÷2⁢x2−x−110⁢x2−3⁢x−1
  3. 112⁢a⁢b⋅50⁢a2⁢(a−b)2a2−b2⋅6⁢ba⁢(a−b)
  4. b2−a2(a−b)2⋅12⁢a⁢(a−b)36⁢a2⁢b⋅9⁢a⁢b⁢(a−b)a+b
  5. x3+y35⁢x⁢y⋅x2−y2x2−2⁢x⁢y+y2⋅25⁢x2⁢y(y+x)2
Answers to odd exercises:
21. 63⁢xx+12;x≠−12,0
23. x−83⁢x⁢(x−4);x≠−8,0,4
25. 5⁢(x+1);x≠−52,0,2		 27. (x+7)25⁢x⁢(x−3);x≠−7,−2,0,3
29. −5⁢x+6x−3;x≠−54,−1,1,3
31. (x+6)⁢(6⁢x−1)(x+3)⁢(2⁢x−9);x≠−3,16,2,92,5		 33. 25a+b;a\new0,±b,b≠0]
35. 5⁢x⁢(x2−x⁢y+y2)x−y;x≠0,±y,y≠0

 ★ Multiply or divide as indicated, state the restrictions, and simplify.

  1. 3⁢x⁢y2(2⁢y+x)2⋅2⁢x2+5⁢x⁢y+2⁢y29⁢x2⋅x3+8⁢y36⁢x⁢y2+3⁢y3
  2. 2⁢x+5x−3⋅x2−95⁢x4÷2⁢x2+15⁢x+2525⁢x5
  3. 5⁢x2−15⁢x9⁢x2−4⋅3⁢x−220⁢x3÷x−33⁢x2−x−2
  4. x2+5⁢x−50x2+5⁢x−14÷x2−25x2−49⋅x−2x2+3⁢x−70
  1. x2−x−564⁢x2−4⁢x−3÷2⁢x2+11⁢x−2125−9⁢x2⋅4⁢x2−12⁢x+93⁢x2−19⁢x−40
  2. 20⁢x2−8⁢x−16⁢x2+13⁢x+6÷1−100⁢x23⁢x2−x−2⋅10⁢x−12⁢x2−3⁢x+1
  3. 12⁢x2−13⁢x+1x2+18⁢x+81÷(144⁢x2−1)⋅x2+14⁢x+4512⁢x2−11⁢x−1
Answers to odd exercises:
   37. 5⁢x⁢(x+3)x+5 ; x≠3,0,−5,−52 39. 1x+5 ;  x≠±7,2,±5,−10
  		 41. −12⁢x+3;  x≠−23,±110,1,12

 

C: Add or Subtract Rational Expressions.

Exercise 0.6.3 

★ Add or subtract and simplify. State the restrictions.

  1. 3⁢x3⁢x+4+23⁢x+4
  2. 3⁢x2⁢x−1−2⁢x+12⁢x−1
  3. x−22⁢x2−11⁢x−6+x+32⁢x2−11⁢x−6
  4. 4⁢x−13⁢x2+2⁢x−5−x−63⁢x2+2⁢x−5
  5. 1x−2⁢x
  6. 4x3−1x
  1. 1x−1−5
  2. 1x+7−1
  3. 1x−2−13⁢x+4
  4. 25⁢x−2−xx+3
  5. 1x2+1x−2
  6. 2⁢xx+2x−2
  1. 3⁢x−7x⁢(x−7)+17−x
  2. 28−x+3⁢x2−1x2⁢(x−8)
  3. x−1x2−25−2x2−10⁢x+25
  4. x+12⁢x2+5⁢x−3−x4⁢x2−1
  5. xx2+4⁢x−2x2+8⁢x+16
  6. 2⁢x−14⁢x2+8⁢x−5−34⁢x2+20⁢x+25
Answers to odd exercises:
51. 3⁢x+23⁢x+4;x≠−43
53. 1x−6;x≠−12,6
55. 1−2⁢x2x;x≠0		 57. −5⁢x+6x−1;x≠1
59. 2⁢(x+3)(x−2)⁢(3⁢x+4);x≠−43,2
61. (x−1)⁢(x+2)x2⁢(x−2);x≠0,2		 63. 2⁢x−7x⁢(x−7);x≠0,7
65. x2−8⁢x−5(x+5)⁢(x−5)2;x≠±5
67. x+2(x+4)2;x≠0,−4

★ Add or subtract and simplify. State the restrictions.

  1. 5−x7⁢x+x2−x+249−x2
  2. 2⁢x4⁢x2+x−x+18⁢x2+6⁢x+1
  3. x−12⁢x2−7⁢x−4+2⁢x−1x2−5⁢x+4
  4. 2⁢(x+3)3⁢x2−5⁢x−2+4−x3⁢x2+10⁢x+3
  5. x24+2⁢x2−2x4+2⁢x2
  6. 3⁢x4⁢x4+6⁢x3−2⁢x26⁢x3+9⁢x2
  1. 3⁢x2−12x4−8⁢x2+16−x2+24−x2
  2. x22⁢x2+1+6⁢x2−242⁢x4−7⁢x2−4
  3. 1+3x−5⁢x−1x2
  4. 4+2x−6⁢x−1x2
  5. 2⁢xx−8−13⁢x+1−2⁢x+93⁢x2−23⁢x−8
  6. 4⁢xx−2−103⁢x+1−19⁢x+183⁢x2−5⁢x−2
  1. 1x−1+1(x−1)2−1x2−1
  2. 1x−2−1x2−4+1(x−2)2
  3. 2⁢x+1x−1−3⁢x2⁢x2−3⁢x+1+x+1x−2⁢x2
  4. 5⁢x22⁢x2+2⁢x−x2−4⁢xx2−2⁢x+4+2⁢x24+2⁢x−2⁢x2
  5. x+22⁢x⁢(3⁢x−2)+4⁢x(x−2)⁢(3⁢x−2)−3⁢x+22⁢x⁢(x−2)
  6. 10⁢xx⁢(x−5)−2⁢x2(2⁢x−5)⁢(x−5)−5⁢xx⁢(2⁢x−5)
Answers to odd exercises:
69. 7⁢(5−2⁢x)x⁢(7+x)⁢(7−x);x≠−7,0,7
71. x⁢(5⁢x−2)(x−4)⁢(x−1)⁢(2⁢x+1);x≠−12,1,4
73. x2−22⁢x2;x≠0		 75. x2+5(x+2)⁢(x−2);x≠±2
77. (x−1)2x2;x≠0
79. 2⁢x−1x−8;x≠−13,8		 81. x2+1(x−1)2⁢(x+1);x≠±1
83. 2⁢x+1x;x≠0,12,1
85. 0;x≠0,23,2

★ Add or subtract and simplify.

  1. x−2+y−2
  2. x−2+(2⁢y)−2
  3. 2⁢x−1+y−2
  4. x−2−4⁢y−1
  1. 16⁢x−2+y2
  2. x⁢y−1−y⁢x−1
  3. 3⁢(x+y)−1+x−2
  4. 2⁢(x−y)−2−(x−y)−1
  1. a−2−(a+b)−1
  2. (a−b)−1−(a+b)−1
  3. x−n+y−n
  4. x⁢y−n+y⁢x−n
Answers to odd exercises:
87. y2+x2x2⁢y2
89. x+2⁢y2x⁢y2		 91. x2⁢y2+16x2
93. 3⁢x2+x+yx2⁢(x+y)		 95. a+b−a2a2⁢(a+b)
97. xn+ynxn⁢yn

 

D: Simplify Complex Rational Expressions.

Exercise 0.6.4 

★ Simplify each complex rational expression.

  1. 75⁢x2(x−3)225⁢x3x−3
  1. x+536⁢x3(x+5)39⁢x2
  1. x2−3632⁢x5x−64⁢x3
  1. x−856⁢x2x2−647⁢x3 
  1. 5⁢x+12⁢x2+x−1025⁢x2+10⁢x+14⁢x2−25
  2. 4⁢x2−27⁢x−74⁢x2−1x−76⁢x2−x−1
  3. x2−4⁢x−52⁢x2+3⁢x+1x2−10⁢x+252⁢x2+7⁢x+3
  4. 5⁢x2+9⁢x−2x2+4⁢x+410⁢x2+3⁢x−14⁢x2+7⁢x−2
  5. x215−3x
  6. 4x−32⁢x2
  1. 13−1x19−1x2
  2. 25+1x425−1x2
  3. 1y2−366−1y
  4. 15−1y1y2−125
  5. 1−6x+8x23−5x−2x2
  6. 2+13x−7x23+1x−10x2
  1. 9−12x+4x29−4x2
  2. 4−25x24−8x−5x2
  3. 1x+53⁢x−123⁢x−1−1x
  4. 2x−5−1x1x−3x−5
  5. 1x+1+2x−22x−3−1x−2
  6. 4x+5−1x−33x−3+12⁢x−1
  1. x−13⁢x−1−1x+1x−1x+1−2x+1
  2. x+13⁢x+5−1x+32x+3−x+1x+3
  3. 2⁢x+32⁢x−3+2⁢x−32⁢x+32⁢x+32⁢x−3−2⁢x−32⁢x+3
  4. x−1x+1−x+1x−1x+1x−1−x−1x+1
  5. 12⁢x+5−12⁢x−5+4⁢x4⁢x2−2512⁢x+5+12⁢x−5+4⁢x4⁢x2−25
  6. 13⁢x−1+13⁢x+13⁢x3⁢x−1−13⁢x+1−6⁢x9⁢x2−1
Answers to odd exercises:
101. 3x⁢(x−3)
103. x+68⁢x2
105. 2⁢x−5(x−2)⁢(5⁢x+1)
107. x+3x−5		 109. 5⁢x3x−15
111. 3⁢xx+3
113. −6⁢y+1y		 115. x−43⁢x+1
117. 3⁢x−23⁢x+2
119. −8⁢x−1x−1		 121. 3⁢x⁢(x−3)(x+1)⁢(x−1)
123. x(3⁢x−1)
125. 4⁢x2+912⁢x
127. 2⁢x−54⁢x

★ Simplify each complex rational expression.

  1. 11+11+1x
  2. 1x1−11+1x
  3. 1y−1x1y2−1x2
  4. 2y+1x4y2−1x2
  1. 125⁢y2−1x21x−15⁢y
  2. 16⁢y2−1x21x−4⁢y
  3. 1b+1a1b3+1a3
  4. 1a−1b1b3−1a3 
  1. xy−yx1y2−2x⁢y+1x2
  2. 2y−5x4⁢x−25⁢y2x
  3. x−1+y−1y−2−x−2
  4. y−2−25⁢x−25⁢x−1−y−1
  5. 1−x−1x−x−1
  1. 16−x−2x−1−4
  2. 1−4⁢x−1−21⁢x−21−2⁢x−1−15⁢x−2
  3. x−1−4⁢(3⁢x2)−13−8⁢x−1+16⁢(3⁢x2)−1
  4. (x−3)−1+2⁢x−1x−1−3⁢(x−3)−1
  5. (4⁢x−5)−1+x−2x−2+(3⁢x−10)−1
Answers to odd exercises:
129. x+12⁢x+1
131. x⁢yx+y		 133. −x+5⁢y5⁢x⁢y
135. a2⁢b2a2−a⁢b+b2		 137. x⁢y⁢(x+y)x−y
139. x⁢yx−y
141. 1x+1		 143. x−7x−5
145. −3⁢(x−2)2⁢x+3

E: Mixed Practice with Rational Expressions.

Exercise 0.6.5 

★ Perform the indicated operations. Simplify the result if possible. State the restrictions. 

  1. 108⁢x312⁢x2
  2. 56⁢x2⁢(x−2)28⁢x⁢(x−2)3
  3. 64−x22⁢x2−15⁢x−8
  4. 3⁢x2+28⁢x+981−x2
  5. x2−255⁢x2⋅10⁢x2−15⁢x2⁢x2+7⁢x−15
  6. 7⁢x2−41⁢x−6(x−7)2⋅49−x2x2+x−42
  7. 28⁢x2⁢(2⁢x−3)4⁢x2−9÷7⁢x4⁢x2−12⁢x+9
  8. x2−10⁢x+24x2−8⁢x+16÷2⁢x2−13⁢x+6x2+2⁢x−24
  9. 5⁢x−6x2−36−4⁢xx2−36
  10. 2x+5⁢x
  1. 5x−5+12⁢x
  2. xx−2+3x+3
  3. 7⁢(x−1)4⁢x2−17⁢x+15−2x−3
  4. 5x−19⁢x+252⁢x2+5⁢x
  5. xx−5−2x−3−5⁢(x−3)x2−8⁢x+15
  6. 3⁢x2⁢x−1−x−4x+4+12⁢(2−x)2⁢x2+7⁢x−4
  7. 1t−1+1(t−1)2−1t2−1
  8. 1t−1−2⁢t−5t2−2⁢t+1−5⁢t2−3⁢t−2(t−1)3
  9. 2⁢x−1+x−2
  10. (x−4)−1−2⁢x−2
  1. 17+1x149−1x2
  2. 1100−1x21x−110
  3. 3x−1x−55x+2−2x
  4. 1−12x+35x21−25x2
  5. x−4⁢x−12−5⁢x−1+2⁢x−2
  6. 8⁢x−1+y−1y−2−64⁢x−2
Answers to odd exercises:
151. 9⁢x;x≠0
153. −x+82⁢x+1;x≠−12,8
155. x−5x;x≠−5,0,32
157. 4⁢x⁢(2⁢x−3)22⁢x+3;x≠±32,0		 159. 1x+6;x≠±6
161. 11⁢x−52⁢x⁢(x−5);x≠0,5
163. −14⁢x−5;x≠54,3
 		 165. x−5x−3;x≠3,5
167. t2+1(t+1)⁢(t−1)2;t≠±1
169. 2⁢x+1x2;x≠0
 		 171. 7⁢xx−7;x≠0,7,−7
173. (x+2)⁢(2⁢x−15)(x−5)⁢(3⁢x−4);x≠−2,0,5,43
175. x⁢(x+2)2⁢x−1⁢d≠0,2,12

08/16/2025

0.06e: Exercises - Rational Expressions is shared under a not declared license and was authored, remixed, and/or curated by LibreTexts.

Support Center

How can we help?